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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_7.html">
\begin{equation}
{\bf x}^{\prime}={\bf A}{\bf x},\tag{6.2.1}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\({\bf A}\)</span> is a constant coefficient <span class="process-math">\(n \times n\)</span> matrix. Seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_7.html">
\begin{equation*}
{\bf x}=\vec{\xi}e^{rt},
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\vec{\xi}\)</span> and <span class="process-math">\(r\)</span> are to be determined. Substituting the solution into (<a href="" class="xref" data-knowl="./knowl/eq7_7.html" title="Equation 6.2.1">(6.2.1)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_7.html">
\begin{equation*}
\vec{\xi} r e^{r t}={\bf A}\, \vec{\xi} e^{r t} \to {\bf A} \,\vec{\xi}=r \vec{\xi} \to ({\bf A}-r {\bf I}) \vec{\xi}={\bf 0},
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\({\bf I}\)</span> is an identity matrix. This is a problem of finding the eigenvalues and eigenvectors for the matrix <span class="process-math">\({\bf A}\text{.}\)</span></p>
<span class="incontext"><a href="sec6_2.html#p-257" class="internal">in-context</a></span>
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